Tóm tắt: A Gradient Ricci soliton (GRS) is a generalization of an Einstein manifold and also plays a key role in the Ricci flow theory,as seen in Perelman’s work on the Poincare conjecture or the differen-tiable sphere theorem by S. Brendle and R. Schoen. In low dimension (n a.That formula, in turn, reveals some interesting connection between the geometry and topology of a GRS. The project involves the Ricci flow, Einstein manifolds, and the solution to the Yamabe conjecture. This is a joint work with Xiaodong Cao.